Open Journal of Bophyscs, 3, 3, 7- http://dx.do.org/.436/ojbphy.3.347 Publshed Onlne October 3 (http://www.scrp.org/journal/ojbphy) An Influence of the Nose on the Imagng Algorthm n the Electrcal Impedance Tomography * Hu Zhang, L Wang, Yongjun Zhou, Peje Zhang, Jxa Wu, Yng L 3 Faculty of Physcs and Electronc Engneerng, Xanyang Normal Unversty, Xanyang, Chna School of Hghway, Chang an Unversty, X an, Chna 3 Department of Cvl, Envronmental and Geomatc Engneerng, Unversty College London, London, UK Emal: xdzhxy@63.com Receved July 4, 3; revsed August 3, 3; accepted September 6, 3 Copyrght 3 Hu Zhang et al. Ths s an open access artcle dstrbuted under the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted. ABSTRACT Electrcal mpedance tomography (EIT) reconstructs the nternal mpedance dstrbuton of the body from electrcal measurements on body surface. The algorthm research s one of the man problems of the EIT. Ths paper presents the MPSO-MNR Algorthm, whch s formed by combnng the Modfed Partcle Swarm Optmzaton (MPSO) wth Modfed Newton-Raphson algorthm (MNR), gves the reconstructon results of certan confguratons and analyzes the nfluence of the nose on the MPSO-MNR algorthm n the EIT. The numercal results show that the MPSO-MNR algorthm can reconstruct the resstvty dstrbuton wthn the certan teratons. Wth the movng of the target to the centre of -D soluton doman and the ncrease of nose, the border of the reconstructon objects becomes vague, and the ftness value and the total error ncrease gradually. Keywords: MPSO-MNR Algorthm; Sgnal-To-Nose Rato; Objectve Functon; EIT. Introducton Electrcal mpedance tomography (EIT) s a non-nvasve magng technque, whch ams to reconstruct mages of nternal electrcal property (conductvty, permttvty and permeablty n some hgh-frequency non-medcal applcatons) dstrbutons and varatons by makng electrcal measurements on the body s surface []. The magng methods are classfed as dynamc or statc magng form. The statc magng method reconstructs the electrcal parameter dstrbuton n the soluton doman. Statc EIT magng problem s solved by transformng the problem nto an objectve functon, and usng the teratve method to acheve the electrcal parameter reconstructon [,3], because t s the nonlnear nverse problem. Algorthm research s one of the man problems of the statc EIT. Based on determnstc and stochastc optmzers, many statc EIT algorthms have been proposed [,3], such as Modfed Newton-Raphson (MNR) algorthm [4], Double Constrant Method (DCM), Layer Strppng Method (LSM), the reconstructon algorthms * Supported by specal foundaton of educaton department of Shaanx provncal government (No. JK89), natonal college s student nnovatve project (No. 74). based on the Partcle Swarm Optmzaton (PSO) [5-7], Genetc algorthm (GA) [8] etc. In recent years, some synthetc algorthms have been proposed, for example, Homotopy-Modfed Newton-Raphson algorthm (H- MNR) [], and MPSO-MNR algorthm [9]. From a computatonal pont-of-vew, the determnstc technques (e.g., MNR methods) are very attractve, for example, MNR algorthm has the fast convergence speed and the calbraton nature. However, the startng tral soluton should be closed enough to the actual soluton. The use of the stochastc genetc algorthms (GA) would, n prncple, avod such a crcumstance. But n the GA, varous numercal parameters must be carefully calbrated and customzed to the applcaton. PSO s a robust stochastc algorthm whch overcomes GA s drawbacks []. Takng nto account the features of PSO and MNR, we combne the advantages of MPSO and MNR algorthm to form the MPSO-MNR algorthm. Its performance and nfluence of the nose on the PSO-MNR are analyzed and dscussed n ths paper. The paper s structured as follows. After the ntroducton of the mathematcal model of the EIT (Secton ), Secton 3 presents the MPSO-MNR algorthm, Secton 4 presents a case study to show the MPSO-MNR algorthm
8 performance and the nfluence of the nose on the reconstructon results. Secton 5 provdes some concludng remarks.. Mathematcal Model The prncple of operaton of EIT s shown n Fgure, where the electrcal current s njected accordng to the certan drven pattern, and the electrcal potental of the body s surface s measured by the electrodes. Based on electromagnetc theory, when an njected electrcal current s at a suffcently low frequency, the EIT problem can be treated as a quas-statc problem. Gven the resstvty dstrbuton nsde the body, the potental satsfes the Laplace s equaton and the boundary condtons as follows []: n n z l l J l n soluton doman, () at measurng electrodes, () at the current njecton electrodes, (3) n at no-njecton electrodes. (4) where s the electrcal resstvty dstrbuton n -D crcular doman, zl s the contact mpedance on l-th electrode, J l s the electrc current densty at l-th electrode respectvely, L s the total electrode numbers. In ths paper, we subdvded the -D crcular soluton doman nto small cells by usng Fnte Element Method (FEM) (see Fgure ). The EIT problem s an ll-posed nverse problem. Gven the electrcal resstvty dstrbuton n the soluton doman and the njected current at the body s surface, the forward problem of the EIT s to solve the potental dstrbuton n the soluton space. The nverse problem s to reconstruct the resstvty dstrbuton by the measurng electrcal potental at the electrodes. Fgure. Prncple of operaton of EIT..5.5.5.5 Fgure. -D crcular model. For the all-electrode current model, the EIT problem can be wrtten as []: b A I, (5) N L where, b R, the elements from to N of the matrx b are the electrcal potental of all FEM node n soluton doman, and from N to N L are the electrode potental. I I, I, IL T s the vector of the njectng electrcal current. The matrx A conssts of the stffness matrx of FEM and the coeffcent matrx of the electrodes. The teratve method s used to reconstruct the resstvty dstrbuton. Gven the ntal value of the MPSO- MNR algorthm at random, to update constantly the resstvty dstrbuton by comparng the changes of the ftness value and the total error untl the end condton s satsfed. 3. MPSO-MNR Algorthm PSO has been shown to be effectve n optmzng dffcult multdmensonal dscontnuous problems n a varety of felds, t s not rgorous demand for the teratve ntal value. MNR has the hgh computatonal effcency when the startng tral soluton s closed to the true value. In vew of the advantages of the PSO and MNR algorthm, the MPSO-MNR algorthm s formed n EIT. The dea of the MPSO-MNR s to produce an ntal value closed to the true value by MPSO, to acqure the resstvty dstrbuton through MNR algorthm. 3.. Modfed Partcle Swarm Optmzaton PSO s a knd of stochastc, evolutonary and global optmzaton algorthm. In ths algorthm, the soluton of each optmzaton problem s a brd n search space, whch s called partcle. All partcles have ts ftness value determned by the optmzaton functon, and each partcle s fly drecton and dstance are determned by ts velocty. The partcle swarm s updated by trackng the personal best ( P ) and global best ( P ) at each teraton b g
9 [6,7,]. In ths paper, MPSO updates the populaton usng the populaton mutaton method, whch s to update some partcles usng (6), and the other partcles track the personal best and global best by usng (7), (8), k x 5. * P g 5. * rrp* rand, (6) k k k k k v k wv cr pb x crpg x, (7) x x v, (8) k k k where w s dynamc nerta weght, rrp s the maxmum emprcal value of the electrcal resstvty n soluton doman, rand s the random number between and. Defnton: Ftness Value Functon F f : F 3.. MNR Algorthm l es t v l v l f L t v l l ρ. (9) Defnton: Objectve Functon f : T f ρ v ρv vρv vρv,() where ρ s the vector of the electrcal resstvty dstrbuton, v, v ρ are the measurng and the calculaton electrcal potental value at the surface electrode, respectvely. The k -th teraton of ρ usng MNR algorthm s where k k k ρ ρ Δρ, () Δ T T k k k k k ρ J J J v v ρ ρ ρ ρ, () v ρ J ρ (3) j s the Jacoban matrx. 3.3. MPSO-MNR Algorthm Flow [9] ) Set up the FEM model n the soluton space. ) Obtan the ntal value of the MNR by the MPSO algorthm. Step a). Intalzaton. Intalze the partcle poston ρ, velocty v, Set the acceleraton constant c, c and nerta constant w. Step b). Calculate and evaluate the partcle s ftness value, compare and obtan the Pb, Pg. Step c). Calculate the partcle s velocty, update the swarm. j Step d). Repeat. 3) Reconstruct by the MNR algorthm. Step a). Usng the numercal result of the MPSO algorthm as the ntal value ρ, to calculate the objectve functon value f ρ. Step b). If the end condton s met, end teraton, else go to step c). k Step c). Calculate objectve functon f ρ. k k k Step d). Calculate ρ ρ Δρ, k k, go to step b). 4. Numercal Smulaton Defnton: (TE) TE m es t m t (4) es where and t are the reconstructon value and the true value of the resstvty n the soluton doman, respectvely. m s the element numbers of the soluton doman. Defnton: Sgnal-to-Nose Rato () log v l n, (5) where v l s the measure potental value at the l -th, n s the samplng nose ampltude. 4.. Sngle Target Smulaton A smulaton target s set n the -D crcular doman (see Fgures 3 and 4), the background resstvty s Ω m, the target s.3 Ω m. The teratve ntal value of the MPSO algorthm s set as the random value between and the emprc value. Accordng to the trend of the ftness value and the total error wth the number of teratons, the economc teratons of the MPSO should be about tmes, MNR algorthm about tmes [9]. In order to compare the reconstructon results n dfferent confguratons and nose levels, the 5-th teraton results of the MPSO s serve as the teratve ntal value of the MNR algorthm, the end condton of the MPSO-MNR algorthm s at 6-th tmes of the MNR teraton n ths paper. The colour scale n Fgures 3 and 4 represents the level of the resstvty. Fgures 3 and 4 are the smulaton targets and the reconstructon results n dfferent nose levels, respectvely. The target n Fgure 4 closes to the centre of the soluton doman compared wth Fgure 3. Comparng Fgures 3 and 4, the numercal results show that the border of the reconstructon object s gettng vague and the background become non-unform wth the target movng to the centre of the crcular doman. Nose affects the magng qualty. Wth the ncrease of the nose, that s the decreasng, the border of the
..4. Table. Ftness value and the total error of the sngle target..8.6.4..5.8.6.4..8.6 Noseless 4 db Experment (Fgure 3) Iteratons (algorthm) Ftness value 5 (MPSO).4.49 6 (MNR).6.44 5 (MPSO).894.39 6 (MNR).585.66.5 Fgure 3. Smulaton model and the reconstructon result of the sngle target. Smulaton target; Reconstructon result (non-nose); Reconstructon result ( = 4 db); Reconstructon result ( = 3 db)..4..8.6.4. 3 db Noseless 4 db 5 (MPSO).48.776 6 (MNR).456.56 Experment (Fgure 4) Iteratons (algorthm) Ftness value 5 (MPSO).47.648 6 (MNR).798.7 5 (MPSO).57.769 6 (MNR).477.67..8.6.4..4..8.6.4. Fgure 4. Smulaton model and the reconstructon result of the sngle target. Smulaton target; Reconstructon result (non-nose); Reconstructon result ( = 4 db); Reconstructon result ( = 3 db). reconstructon object become vague. The reconstructon qualty of the EIT s evaluated by the ftness value and the total error. Table gves the ftness value and the total error of the sngle target n dfferent poston and under dfferent nose levels, whch the number of the MPSO teratons are at 5-th tmes and MNR at 6-th tmes. Comparng the ftness value and the.5.5..8.6.4. 3 db 5 (MPSO).935.838 6 (MNR).59.43 total error n dfferent condtons, the results show that the ftness value and the total error ncrease gradually at same teratons wth decreasng of the and the target close to the centre of the soluton doman, whch means the lower magng qualty. Ths result s consstent wth the results of the Fgures 3 and 4. 4.. Two Targets Smulaton The two targets smulaton s presented n ths secton. In the -D crcular doman, the background resstvty s Ω m, and the target s.3 Ω m. The settng of the teratve ntal value and the end condton s same as the settng of the sngle target smulaton. Fgure 5 shows the smulaton targets and the reconstructon results under dfferent nose levels. The ftness value and the total error are presented n Table. The results show that the reconstructon targets border become vague the ftness value and the total error ncrease gradually wth the ncrease of nose. 5. Conclusons Algorthm research s an mportant ssue n EIT technque. The MPSO-MNR algorthm s formed by the MPSO algorthm and the MNR algorthm. It draws the advantages of loosng the ntal value demand and the feature of the global convergence n MPSO algorthm,
..8.6 works n EIT as well..8.4. 6. Acknowledgements.6.4..8.6.4. The authors would lke to thank the revewers whose thoughtful comments added to the effcacy of ths paper. Ths work was supported by foundaton of educaton department of Shaanx provncal government (No. JK89) and natonal college s student nnovatve project (No. 74)..4..8.6.4. Fgure 5. Smulaton experment model and the reconstructon result of the two targets. Smulaton target; Reconstructon result (non-nose); Reconstructon result ( = 4 db); Reconstructon result ( = 3 db). Table. Ftness value and the total error of the two targets. Noseless 4 db 3 db Iteratons (algorthm) Ftness value 5 (MPSO).64.75 6 (MNR)..53 5 (MPSO).36.5 6 (MNR).8.67 5 (MPSO).83.48 6 (MNR).935.339 avods the demand of the ntal value close to the true value and uses the advantages of the fast convergence speed of MNR algorthm. The numercal results show that the MPSO-MNR algorthm can reflect the resstvty dstrbuton n the soluton space wthn the certan teratons. Wth the target movng to the centre of -D soluton doman and the ncreasng of the nose, the border of the reconstructon objects becomes vague, and the ftness value and the total error ncrease gradually. Ths suggests that the nose and the relatve poston of target nfluence the mage qualty. Model degeneracy s a general dffculty n optmzng problems. In ths paper, the resstvty dstrbuton of the certan confguraton can be reconstructed by MPSO- MNR algorthm and the results show changes n performance under dfferent nose levels. Therefore, the ant-nose algorthm wll be the research drecton n the practcal applcaton of the EIT. In addton, the effect of o ther confguratons on MPSO-MNR wll be the research.8.6.4. REFERENCES [] H. Dehghan and M. Soleman, Numercal Modelng Errors n Electrcal Impedance Tomography, Physologcal Measurement, Vol. 8, No. 7, 7, pp. S45-S55. http://dx.do.org/.88/967-3334/8/7/s4 [] G.-Z. Xu, Y. L, et al, Electrcal Impedance Tomography n Bomedcal Engneerng, Chna Machne Press, Bejng,. [3] W. He, C.-Y. Luo, Z. Xu, et al., Prncple of the Electrcal Impedance Tomography, Scence Press, Bejng, 9. [4] Y. L, G. Z. Xu, L. Y. Rao, et al., MNR Method wth Self-Determned Regularzaton Parameters for Solvng Inverse Resstvty Problem, Proceedngs of 7th Annual Internatonal Conference of the IEEE Engneerng n Medcne and Bology, Vol. 3, 5, pp. 653-655. [5] H. Zhang, Y.-J. Zhou and X.-D. Zhang, Algorthm Study of -D Electrcal Impedance Tomography, Journal of Xanyang Normal Unversty, Vol. 7, No.,, pp. 7-9. [6] H. Zhang, X.-D. Zhang, et al., Electromagnetc Imagng of the -D Meda Based on Partcle Swarm Algorthm, Sxth Internatonal Conference on Natural Computaton (ICNC), - August, pp. 6-64. [7] H. Zhang, H. B. Wang, Y. J. Zhou and X. D. Zhang, Research of Electrcal Impedance Tomography Based on the Modfed Partcle Swarm Optmzaton, Eghth Internatonal Conference on Natural Computaton (ICNC), 9-3 May, pp. 7-9. [8] R. Olm, M. Bn and S. Pror, A Genetc Algorthm Approach to Image Reconstructon n Electrcal Impedance Tomography, IEEE Transactons on Evolutonary Computaton, Vol. 4, No.,, pp. 83-88. http://dx.do.org/.9/435.843497 [9] H. Zhang, Y. L, X. M. Wang and X. D. Zhang, MPSO- MNR Algorthm Study of -D Electrcal Impedance Tomography, Computer Engneerng and Applcatons, Vol. 49, No. 3, 3, pp. 9-3. [] M. Donell and A. Massa, Computatonal Approach Based on a Partcle Swarm Optmzer for Mcrowave Imagng of Two-Dmensonal Delectrc Scatterers, IEEE Transactons on Mcrowave Theory and Technques, Vol. 53, No. 5, 5, pp. 76-776. http://dx.do.org/.9/tmtt.5.84768 [] S. Huang, Regularzaton Algorthm Research of Statc Electrcal Impedance Tomography, Ph.D. Dssertaton, Chongqng Unversty, Chongqng, 5.